If the system of linear equations : \( \begin{aligned} & x+y+2 z=6 \\ & 2 x+3 y+a z=a+1 \\ & -x-3 y+b z=2 b …

\(\text{using cramer's rule :}\\ \Delta =\left|\begin{matrix}1 & 1 & 2 \\ 2 & 3 & \mathrm{a} \\ -1 & -3 & \mathrm{b}\end{matrix}\right|=0\\ \Rightarrow 2\mathrm{a}+\mathrm{b}-6=0...\text{(1)}\\ {\Delta }_{3}=\left|\begin{matrix}1 & 1 & 6 \\ 2 & 3 & a+1 \\ -1 & -3 & 2b\end{matrix}\right|=0\\ \Rightarrow \mathrm{a}+\mathrm{b}-8=0...\text{(2)}\\ \text{from eqn}\left(1\right)\text{and}\left(2\right)\\ a=-2,b=10\\ \Rightarrow 7a+3b=16\)